As explained in [1]}, we can extend (REF ) to a morphism of real cone stacks (i.e. geometric stacks over the category \(\operatorname{\bf {PC}}\)  of (non necessarily rational) polyhedral cones):
\(\Phi ^{trop, \mathbb {R}}:\mathcal {J}^{trop, \mathbb {R}}_{g,n}\longrightarrow \mathcal {M}^{trop, \mathbb {R}}_{g,n},\) 
